BASIC CONCEPTS OF ENRICHED CATEGORY THEORY

By G. M. Kelly, T. Bartels, William Boshuck, Maria Manuel Clementino, R. Dawson, Stephen Lack, Tom Leinster, F. Marmolejo, Shane O 'conchuir, C. Pastro and 4 more. Elements of ∞-Category Theory, 2022

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Although numerous contributions from divers authors, over the past fifteen years or so, have brought enriched category theory to a developed state, there is still no connected account of the theory, or even of a substantial part of it. As the applications of the theory continue to expand - some recent examples are given below - the lack of such an account is the more acutely felt. The present book is designed to supply the want in part, by giving a fairly complete treatment of the limited area to which the title refers. The basic concepts of category theory certainly include the notion of functor-category, of limit and colimit, of Kan extension, and of density; with their applications to completions, perhaps including those relative completions given by categories of algebras for limit-defined theories. If we read 'V-category' for 'category' here, this is essentially the list of our chapter-headings below, after the first chapter introducing V-categories. In fact our scope is wider than this might suggest; for what we give is also a selfcontained account of basic category theory as described above, assuming as prior knowledge only the most elementary categorical concepts, and treating the ordinary and enriched cases together from Chapter 3 on.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c59eb75662741d9ed86141bf3dfab6c2ea33c2fc

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Citations in the corpus

Homotopy coherent adjunctions and the formal theory of monads (2013, neutral): “A more thorough treatment can be found in [14] or [22].”
Enriched Categories, Internal Categories and Change of Base (2011, background): “we enrich these basic maps by asking that they respect composites of proarrows in an (op)lax- or pseudo- sense, and generalise Kelly’s doctrinal adjunction results [29] to induce such structures across biadjunctions.”
Higher-dimensional categories: an illustrated guide book (2004, background): “There is a well-developed theory of enrichment [57], but unfortunately this theory is strict and so only gives us strict n-categories.”
Algebraic Operations and Generic Effects (2003, methodology): “We Þrst need to explain an enriched version of the Yoneda lemma as in [ 10 ].”
Semantics for probabilistic programming: higher-order functions, continuous distributions, and soft constraints (2016, neutral): “F ( G ) is the left Kan extension of G along F , see e.g. [ 15].”
Commutativity (2015, background): “A monoid in a monoidal category V is equally well a one-object V-enriched category [31], and the meaning of our theory of commutativity may be clarified by generalising it from V-monoids to V-categories; the basic notion of commuting cospan (1.”
Global Homotopy Theory (2018, background): “For more background on symmetric monoidal categories and enriched category theory we refer the reader to [84].”